Minesweeper is NP-Complete

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Minesweeper is NP-Complete
Notes by Melissa Gymrek Based on a paper by
Richard Kayes 2000
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Minesweeper

• Reducing 3SAT to generalized Minesweeper
• Reducing cSAT to well-know version of Minesweeper

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General Minesweeper
MINESWEEPER G, ? G is a graph and ? is a
partial integer labeling of G, and G can be
filled with mines in such a way that any node v
labeled m has exactly m neighboring nodes
containing mines. Deciding if a graph is in the
MINESWEEPER language is NP-complete - Polynomial
time verification - Reduce from 3SAT in
polynomial time
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Polynomial Time Verification

• For each node v labeled m
• Check that exactly m neighbors contain mines
• O(E) time clearly polynomial

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Reduce from 3SAT
3SAT instance w
MINESW instance Z
f(w)

• Function f converts a 3SAT instance to a MINESW
  instance in polynomial time
• Z is satisfiable iff w is satisfiable

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3SAT Review
Boolean 3CNF formula (A V B V C) (A V D V
C) N variables (A, B, C, D) in this
instance M clauses (here 2 clauses are
shown) Question Is this boolean formula
satisfiable?
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3SAT ? MINESWEEPER
1
0
0
xi
xi
Make a gadget for each variable
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3SAT ? MINESWEEPER
For clause (A V B V C)
N
0
0
0
N-1 satellite nodes
...
N
Connect to variable gadgets
0
0
0
0
0
0
0
0
0
1
1
1
A
B
A
C
Make a gadget for each clause
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3SAT ? MINESWEEPER

• Conversion took polynomial time
• 1 gadget for each of the N vars O(N)
• 1 gadget for each of M clauses O(MN)
• Total O(N(M1)) time

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Minesweeper as we know it
MINESWEEPER Problem Given a rectangular grid
partially marked with numbers and/or mines, some
squares being left blank, determine whether there
is some pattern of mines in the blank squares
giving rise to the numbers seen. Deciding if a
graph is in the MINESWEEPER language is
NP-complete - Polynomial time verification -
Reduce from cSAT in polynomial time
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Wire
Either all the x's or all the x''s are mines. If
it is the x's, we call it true, if the x''s, we
call it false
http//www.claymath.org/Popular_Lectures/Minesweep
er/msweep-fig3.jpg
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Manipulating Wires
Minesweeper is NP-Complete, Kayes
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Manipulating Wires
Minesweeper is NP-Complete, Kayes
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NOT gate
Inverts the sign of a wire
http//www.claymath.org/Popular_Lectures/Minesweep
er/msweep-fig3.jpg
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More gates

• We can now manipulate/invert wires
• Cross wires? First make planar XOR, then use XOR
  and three way splitter to cross wires
• We have NOT, and AND, universal!

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More gates
Minesweeper is NP-Complete, Kayes
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AND gate
Minesweeper is NP-Complete, Kayes
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NAND is universal!

• (A nand A) nand (B nand B) A v B
• (A nand B) nand (A nand B) A B
• (A nand A) A

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Tetris is NP-complete
Ron Breukelaar, Erik Demaine, Susan Hohenberger,
Hendrik Jan Hoogeboom, Walter Kosters,
David Liben-Nowell published 2004
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Initial Board
T lock
T notches(target sum)
(it is possible toactually get here)
s columns (one per sum)
bane
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Piece Sequence

• For each input ai

(ai reps)
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Failure to Launch
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Forced Moves
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FinalePieces
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FinalePieces
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FinalePieces
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FinalePieces
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FinalePieces
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FinalePieces
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Summary

• If theres a 3-partition, can win TetrisGet
  tons of lines, Tetrises, live forever, etc.
• If theres no 3-partition, must lose TetrisDie,
  no lines, no Tetrises, etc.

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Open Problems

• What if the initial board is empty?
• What about Tetris with O(1) columns?
• What about Tetris with O(1) rows?
• What about restricted piece sets (e.g. just )?
• What if every move drops from high up(no
  last-minute slides)?
• Is two-player Tetris PSPACE-complete?
• What can we say about online (regular) Tetris?